chiang ch5b

8/12/2019 Chiang Ch5b

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Ch. 5b Linear Models & Matrix

Algebra5.5 Cramer's Rule

5.6 Application to Market and National-Income Models

5.7 Leontief Input-Output Models5.8 Limitations of Static Analysis

1


2/33

5.2 Evaluating a third-order determinant

Evaluating a 3rd order determinant by Laplace expansion

2

ijjiiji

ii MCCaA

CaCaCaA

aa

aaa

aa

aaa

aa

aaaA

aaa

aaa

aaa

A

1where

1columndownExpanding

AfindGiven

3

1

11

313121211111

2322

131231

3332

131221

3332

232211

333231

232221

131211

Ch. 5a Linear Models and Matrix Algebra5.1 - 5.4


3/33

5.5 Deriving Cramers Rule(nxn)

dAA

x

d

d

d

CCC

CCC

CCC

A

x

x

x

dAA

x

nx

nnxn

nnnn

n

n

x

nx

n

*adjoint1

find

tion,multiplicaoflaweassociativtheApplying

1

adjoint1

*

1

2

1

21

22212

12111

11

1

*

*

*

*

2

1

Ch. 5b Linear Models and Matrix Algebra5.5 - 5.8 3


4/33

5.5 Deriving Cramers Rule(3x3)

AAx

aad

aad

aad

A

Aaa

aad

aa

aad

aa

aadCd

MCCdCdCdCd

Cd

Cd

Cd

ACdCdCd

CdCdCd

CdCdCd

Ax

x

x

*

i

ii

ijji

iji

i

ii

iii

i

ii

i

ii

111

33323

23222

13121

1

12322

13123

3332

13122

3332

23221

3

1

1

31212111

3

1

1

3

13

3

1

2

3

1

1

333232131

323222121

313212111

*3

*2

*1

such thatAFind.4)

)3

1 where)2

11)1



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AA

aaa

aaa

aaaaad

aad

aad

Ca

Cd

x

i

ii

i

ii

1

333231

232221

131211

33323

23222

13121

3

1

11

3

1

1

*1


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AA

aaa

aaa

aaaada

ada

ada

Ca

Cd

x

i

ii

i

ii

2

333231

232221

131211

33331

23221

13111

3

1

22

3

1

2

*2


7/33

5.5 Deriving Cramers Rule

(3x3)


AA

aaa

aaa

aaadaa

daa

daa

Ca

Cd

x

i

ii

i

ii

3

333231

232221

131211

33231

22221

11211

3

1

33

3

1

3

*3


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A

A

Ca

Cd

x

A

A

Ca

Cd

x

A

A

Ca

Cd

x

n

n

i

inin

n

i

ini

n

n

i

ii

n

i

ii

n

i

ii

n

i

ii

1

1*

2

1

22

1

2*

2

1

1

11

1

1*

1


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n

i

inin

n

i

ini

n

n

i

inin

n

i

inin

n

iii

n

i

iin

i

ii

n

i

ii

n

i

ii

n

i

iin

i

ii

n

i

ii

n

i

ijij

Ca

Cd

xCaACdA

x

Ca

Cd

xCaACdA

x

Ca

Cd

xCaACdA

x

scalarCaA

1

1*th

11

*

122

1

2*

2

nd

1

22

1

2

*

2

1

11

1

1*

1

st

1

11

1

1

*

1

1

exp.col.nfor the1

exp.col.2for the1

exp.col.1for the1

expansioncolumnaforj)(jwhere:patterna

Given thatforLooking


10/33


AA

AA

AA

AA

Cd

Cd

Cd

Cd

A

x

x

x

x

nn

i

ini

n

i

ii

n

iii

n

i

ii

n

3

2

1

1

1

3

12

1

1

*

*

*

*

1

3

2

1



11/33

5.6 Applications to Market and National-income

Models: Matrix Inversions

65z3;1y;21

3

44

312-

61014-3-3-6

6

1

z

yx

312-

61014-

3-3-6

6

1A

3

4

4

d;

z

y

x

x;

301

125

214

InversionMatrix

1-

1

x

dAx

A

dAx

65

31

21

;6

;5;2;3

3

4

4

d;

z

y

x

x;

301

125

214

RulesCramer'

3

2

1

321

AAz

AAy

AAx

A

AAA

A

dAx



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5.6 Macro model

Section 3.5, Exercise 3.5-2 (a-d), p. 47 and

Section 5.6, Exercise 5.6-2 (a-b), p. 111

Given the following model

(a) Identify the endogenous variables

(b) Give the economic meaning of the parameter g

(c) Find the equilibrium national income

(substitution)(d) What restriction on the parameters is needed

for a solution to exist?

Find Y, C, G by (a) matrix inversion (b) Cramers

rule Ch. 5b Linear Models and Matrix Algebra5.5 - 5.8 12


13/33


14/33

5.6 The macro model (3.5-2, p.47)


gbg

bA

1

10

01111

tdeterminanThe

010

01

111formMatrix

0

0

bTa

I

G

C

Y

g

b


15/33

5.6 Application to Market & National Income

Models: Cramers rule (3.5-2, p. 47)


)(1

00

1

11

)(1

11

10

0

11

)(1100

01

11

constantsoftorcolumn vecwith the

tdeterminantheofcolumnjththereplacingbyxofsolutiontheFinding

00*

000

0

00*

000

0

00*

000

0

*

j

gb

IbTag

A

AGIbTag

g

bTab

I

A

gb

bTagbI

A

ACbTagbI

g

bTab

I

A

gb

bTaI

A

AYbTaIbTa

I

A

G

G

C

C

Y

Y


16/33


Models: Matrix Inversion (3.5-2, p. 47)


gbg

bA

1

10

01

111

10

01

111

g

bA

bb

gg

gb

C

11

11

1

bgg

bgbCT

1

1

111


17/33


Models: Matrix Inversion (3.5-2, p. 47)


gbbTaIg

G

gb

bTagbI

C

gb

bTaIY

1

1

1

1

00*

00*

00*

01

1

111

1

10

0

*

*

*

bTa

I

bgg

bgbgb

G

C

Y


18/33


5.7 Leontief Input-Output ModelsMiller and Blair 2-3, Table 2-3,

p 15 Economic Flows ($ millions)

Inputs (cols)

Outputs (rows)

Sector 1

(zi1)

Sector 2

(zi2)

Final demand

(di)

Total gross

output

(xi)Intermediate

inputs: Sector 1150 500 350 1000

Intermediate

inputs: Sector 2200 100 1700 2000

Primary inputs

(wi)650 1400 1100 3150

Total outlays

(xi)1000 2000 3150 6150


19/33


5.7 Leontief Input-Output ModelsMiller and Blair 2-3, Table 2-3,

p 15 Inter-industry flows as factor shares

Inputs (cols)

Outputs (rows)

Sector 1

(zi1/x1

=ai1)

Sector 2

(zi2/x2

=ai2)

Final demand

(di)

Total output

(xi)Intermediate

inputs: Sector 10.15 0.25 350 1000

Intermediate

inputs: Sector 20.20 0.05 1700 2000

Primary inputs(wi/xi)0.65 0.70 1100 3150

Total outlays

(xi/xi)1.00 1.00 3150 6150


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21/33

5.7 Leontief Input-Output ModelsStructure of an input-output model

nnnnnnn

nn

nn

dxaxaxax

dxaxaxax

dxaxaxax

2211

222221212

112121111

formstandardThe


17002000*05.01000*20.020002sector

3502000*25.01000*15.010001sector

daa x 2i21i1i

xx


22/33



nnnnnn

n

n

d

d

d

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

1

1

1

formMatrix

1700

350

2000

1000

05.0120.0

25.015.01

2sector

1sector


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nnnnnn

n

n

d

d

d

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211

100

010

001

decomposedformmatrixThe

1700

350

2000

1000

05.020.0

25.015.0

10

01

dxAI


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1221126400

3300025411

050200

250150

050200

250150

10

01

analogyby

algebrascalar1xif11

2(a))-9.5250,p.t,Wainwrigh&(ChiangionapproximatseriesTaylor

2

12

0

1

12

0

1

00111

..

.....

..

..

..

..

A...AAIAAI

x...xxxx

IAAAAIAA

n

n

n

n

n

n

-


Miller & Blair, p. 102


25/33


26/33


1700

350

1221.12640.0

3300.02541.1

2000

1000

1700

350

85.020.0

25.095.0

7575.0

1

2000

1000

7575.0)25.0*20.0()095.*85.0(

1700

350

95.020.0

25.085.0

2000

1000

matrixtheInverting

1

1*

AI

dAIx



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2000208170092outputbase100056189350

20001700*1221.1350*2640.0

10001700*3300.0350*2541.1

1700

350

1221.12640.0

3300.02541.1

inversionmatrixwithresultstheChecking

*2

*1

*2

*1

*

*

2

1

x

x

x

x

x

x



28/33



20007575.0

1515

7575.0

170020.0

35085.0

10007575.0

5.757

7575.0

95.01700

25.0350

7575.0;1700

350

95.020.0

25.085.0

rulesCramer'withresultstheChecking

2*

2

1*

1

1

*

2

*

1

AI

AIx

AI

AIx

AIx

x


29/33


30/33


209220817001841439561178700

20921700*1221.1700*2640.0

14391700*3300.0700*2541.1

1700

700

1221.12640.0

3300.02541.1

in total?needwedoinputprimarymuchhow

1,sectorfromnconsumptiodoubleweIf

*

*

*

*

*

*

2

1

2

1

2

1

x

x

x

x

x

x



31/33


multiplieroutputs1'sector52.100.1$

52.1$52.1$264.0$2541.1$

2sectorin264.0$0*1221.11*2640.0

1sectorin2541.1$0*3300.01*2541.1

0

1

1221.12640.0

3300.02541.1

economy?in theproductionincreasewedomuchhow

$1,by1sectorfromnconsumptioincreaseweIf

*

*

*

*

2

1

2

1

x

x

x

x



32/33

5.8 Limitations of Static Analysis

Static analysis solves for the

endogenous variables for one

equilibrium Comparative statics show the shifts

between equilibriums

Dynamics analysis looks at the

attainability and stability of theequilibrium



33/33

5.6 Application to Market and National-Income ModelsMarket model

National-income model

Matrix algebra vs. elimination of variables

Why use matrix method at all?

Compact notation

Test existence of a unique solution Handy solution expressions subject to

manipulation

Ch 5b Linear Models and Matrix Algebra

chiang ch5b

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